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Intlab  Interval Laboratory
"... . INTLAB is a Matlab toolbox supporting real and complex interval scalars, vectors, and matrices, as well as sparse real and complex interval matrices. It is designed to be very fast. In fact, it is not much slower than the fastest pure floating point algorithms using the fastest compilers available ..."
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Cited by 134 (12 self)
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. INTLAB is a Matlab toolbox supporting real and complex interval scalars, vectors, and matrices, as well as sparse real and complex interval matrices. It is designed to be very fast. In fact, it is not much slower than the fastest pure floating point algorithms using the fastest compilers available (the latter, of course, without verification of the result). Portability is assured by implementing all algorithms in Matlab itself with exception of exactly three routines for switching the rounding downwards, upwards and to nearest. Timing comparisons show that the used concept achieves the anticipated speed with identical code on a variety of computers, ranging from PC's to parallel computers. INTLAB may be freely copied from our home page. 1. Introduction. The INTLAB concept splits into two parts. First, a new concept of a fast interval library is introduced. The main advantage (and difference to existing interval libraries) is that identical code can be used on a variety of computer a...
Interval Computations: Introduction, Uses, and Resources
 Euromath Bulletin
, 1996
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On directed interval arithmetic and its applications
, 1995
"... We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction ..."
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Cited by 20 (4 self)
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We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction of special notations and a socalled normal form of directed intervals. As an application, it has been shown that both interval systems can be used for the computation of tight inner and outer inclusions of ranges of functions and consequently for the development of software for automatic computation of ranges of functions.
Using Directed Acyclic Graphs to Coordinate Propagation and Search for Numerical Constraint Satisfaction Problems
 In Proceedings of the 16th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2004
, 2004
"... A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) ..."
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Cited by 17 (6 self)
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A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) enabling the flexible choice of the interval inclusion functions during propagation. We then propose a new simple algorithm which coordinates constraint propagation and exhaustive search for solving numerical constraint satisfaction problems. The experiments carried out on different problems show that the new approach outperforms previously available propagation techniques by an order of magnitude or more in speed, while being roughly the same quality w.r.t. enclosure properties. I.
Ten methods to bound multiple roots of polynomials
 J. Comput. Appl. Math. (JCAM
"... Abstract. Given a univariate polynomial P with a kfold multiple root or a kfold root cluster near some z̃, we discuss various different methods to compute a disc near z ̃ which either contains exactly or contains at least k roots of P. Many of the presented methods are known, some are new. We are ..."
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Abstract. Given a univariate polynomial P with a kfold multiple root or a kfold root cluster near some z̃, we discuss various different methods to compute a disc near z ̃ which either contains exactly or contains at least k roots of P. Many of the presented methods are known, some are new. We are especially interested in rigorous methods, that is taking into account all possible effects of rounding errors. In other words every computed bound for a root cluster shall be mathematically correct. We display extensive test sets comparing the methods under different circumstances. Based on the results we present a hybrid method combining five of the previous methods which, for given z̃, i) detects the number k of roots near z ̃ and ii) computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster. Therefore, the resulting discs are numerically nearly optimal. 1. Introduction and notation. Throughout the paper denote by P = n∑ ν=0 pνz ν ∈ C[z] a (real or
Interval Computations and IntervalRelated Statistical Techniques: Tools for Estimating Uncertainty of the Results of Data Processing and Indirect Measurements
"... In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on ..."
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Cited by 15 (9 self)
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In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on [−∆, ∆], and to use the corresponding statistical techniques. In some situations, however, this approach underestimates the error of indirect measurements. It is therefore desirable to directly process this interval uncertainty. Such “interval computations” methods have been developed since the 1950s. In this chapter, we provide a brief overview of related algorithms, results, and remaining open problems.
Parameter reconstruction for biochemical networks using interval analysis
 Reliab. Comput
"... Abstract. In recent years, the modeling and simulation of biochemical networks has attracted increasing attention. Such networks are commonly modeled by systems of ordinary differential equations, a special class of which are known as Ssystems. These systems are specifically designed to mimic kine ..."
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Cited by 6 (0 self)
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Abstract. In recent years, the modeling and simulation of biochemical networks has attracted increasing attention. Such networks are commonly modeled by systems of ordinary differential equations, a special class of which are known as Ssystems. These systems are specifically designed to mimic kinetic reactions, and are sufficiently general to model genetic networks, metabolic networks, and signal transduction cascades. The parameters of an Ssystem correspond to various kinetic rates of the underlying reactions, and one of the main challenges is to determine approximate values of these parameters, given measured (or simulated) time traces of the involved reactants. Due to the high dimensionality of the problem, a straightforward optimization strategy will rarely produce correct parameter values. Instead, almost all methods available utilize genetic/evolutionary algorithms to perform the nonlinear parameter fitting. We propose a completely deterministic approach, which is based on interval analysis. This allows us to examine entire sets of parameters, and thus to exhaust the global search within a finite number of steps. The proposed method can in principle be applied to any system of finitely parameterized differential equations, and, as we demonstrate, yields encouraging results for low dimensional Ssystems. 1.
Safe motion planning computation for databasing balanced movement of humanoid robots
 in ICRA 2009
"... Abstract — Motion databasing is an important topic in robotics research. Humanoid robots have a large number of degrees of freedom and their motions have to satisfy a set of constraints (balance, maximal joint torque velocity and angle values). Thus motion planning cannot efficiently be done online. ..."
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Abstract — Motion databasing is an important topic in robotics research. Humanoid robots have a large number of degrees of freedom and their motions have to satisfy a set of constraints (balance, maximal joint torque velocity and angle values). Thus motion planning cannot efficiently be done online. The computation of optimal motions is performed offline to create databases that transform the problem of large computation time into a problem of large memory space. Motion planning can be seen as a SemiInfinite Programming problem (SIP) since it involves a finite number of variables over an infinite set of constraints. Most methods solve the SIP problem by transforming it into a finite programming one using a discretization over a prescribed grid. We show that this approach is risky because it can lead to motions which may violate one or several constraints. Then we introduce our new method for planning safe motions. It uses Interval Analysis techniques in order to achieve a safe discretization of the constraints. We show how to implement this method and use it with stateoftheart constrained optimization packages. Then, we illustrate its capabilities for planning safe motions dedicated to the HOAP3 humanoid robot.
Diagrammatic representation for interval arithmetic
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 2001
"... The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation ..."
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Cited by 4 (0 self)
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The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation devised earlier by the author to represent interval relations. First, the MRdiagram is defined together with appropriate graphical notions and constructions for basic interval relations and operations. Second, diagrammatic constructions for all standard arithmetic operations are presented. Several examples of the use of these constructions to aid reasoning about various simple, though nontrivial, properties of interval arithmetic are included in order to show how the representation facilitates both deeper understanding of the subject matter and reasoning about its properties.